Simulations of non homogeneous viscous flows with incompressibility constraints

Abstract

This presentation is an overview on the development of numerical methods for the simulation of non homogeneous flows with incompressibility constraints. We are particularly interested in systems of partial differential equations describing certain mixture flows, like the Kazhikhov-Smagulov system which can be used to model powder-snow avalanches. It turns out that the Incompressible Navier-Stokes system with variable density is a relevant step towards the treatment of such models, and it allows us to bring out some interesting numerical difficulties. We should handle equations of different types, roughly speaking transport and diffusion equations. We present two strategies based on time-splitting. The former relies on a hybrid approach, coupling finite volume and finite element methods. The latter extends discrete duality finite volume schemes for such non homogeneous flows. The methods are confronted to exact solutions and to the simulation of Rayleigh-Taylor instabilities.

An explicit finite volume scheme on staggered grids for the Euler equations: Unstructured meshes, stability analysis, and energy conservation

“…Applying (11) in Lemma 3.3 shows that ( 42) is satisfied. Finally, we define a conservative flux of total energy T D σ,σ * ,s through the interface s of the diamond cell D σ,σ * by…”

“…Applying Lemma 3.3, on each primal cell C = K with X K,σ = K K,σ provides a function ω K which satisfies (11) and (12). For the dual mesh, we proceed as for the primal mesh to define ω K * from the fluxes X K * ,σ * = K K * ,σ * on the interfaces of K * .…”

“…12 provides an example of the direct and the barycentric approaches, starting from the same structured triangular primal mesh. Clearly, the barycentric dual mesh can have a much more complicated structure than the direct dual mesh, requiring more involved descriptors in the code, but in practice it can be more robust with respect to mesh orientation effects and stability issues, see for instance [10,11]. It might happen that the direct dual mesh produces non convex diamonds cells, in which case the edges of the dual mesh are not included in the diamond.…”

A combined finite volumes ‐ finite elements method for a low‐Mach model

“…Similarly to other references 27,40 or to our previous contributions for incompressible fluids, 35,37,41 an extrapolation was used to approximate the convective velocity at time t n + 1 by 2u n − u n − 1 in (24) and (27). Moreover, the velocity used in (20) and (21) is extrapolated at time (t n + t n + 1 )∕2 as indicated in (22), which is necessary to reach the second-order accuracy.…”